Grill sausages, covered, over medium heat until a thermometer reads 160°, 10-12 minutes, turning occasionally. Cut into 1/4-in. slices., Brush both sides of flatbreads with oil. Grill flatbreads, covered, over medium heat until bottoms are lightly browned, 2-3 minutes., Remove from grill. Layer grilled sides with sauce, sausage, cheeses and basil. Return to grill; cook, covered, until cheese is melted, 2-3 minutes longer.